To some extent,
the partial determinants share some common properties relative to partial traces.
For instance,
it is easy to see that if \({ H} \in \mathbb {M}_n(\mathbb {M}_k)\)  is positive semidefinite, then both \(\mathrm {det}_1 H\)  and \(\mathrm {det}_2 H\)  are positive semidefinite; see, e.g. [1]}.
Moreover, it was proved in [2]} that
\( \mathrm {det} ({\rm tr}_1 H) \ge {\rm tr}(\mathrm {det}_2 H) ,\) 
